Atomic Propositions Research Articles

Model checking dynamic pushdown networks

Abstract A dynamic pushdown network (DPN) is a set of pushdown systems (PDSs) where each process can dynamically create new instances of PDSs. DPNs are a natural model of multi-threaded programs with (possibly recursive) procedure calls and thread creation. Thus, it is important to have model checking algorithms for DPNs. We consider in this work model checking DPNs against single-indexed LTL and CTL properties of the form ⋀ f i such that f i is a LTL/CTL formula over the PDS i . We consider the model checking problems w.r.t. simple valuations (i.e., whether a configuration satisfies an atomic proposition depends only on its control location) and w.r.t. regular valuations (i.e., the set of the configurations satisfying an atomic proposition is a regular set of configurations). We show that these model checking problems are decidable. We propose automata-based approaches for computing the set of configurations of a DPN that satisfy the corresponding single-indexed LTL/CTL formula.

Automatically inferring loop invariants via algorithmic learning

By combining algorithmic learning, decision procedures, predicate abstraction and simple templates for quantified formulae, we present an automated technique for finding loop invariants. Theoretically, this technique can find arbitrary first-order invariants (modulo a fixed set of atomic propositions and an underlying satisfiability modulo theories solver) in the form of the given template and exploit the flexibility in invariants by a simple randomized mechanism. In our study, the proposed technique was able to find quantified invariants for loops from the Linux source and other realistic programs. Our contribution is a simpler technique than the previous works yet with a reasonable derivation power.

Efficient CTL model-checking for pushdown systems

Pushdown systems (PDS) are well adapted to model sequential programs with (possibly recursive) procedure calls. Therefore, it is important to have efficient model checking algorithms for PDSs. We consider in this paper CTL model checking for PDSs. We consider the “standard” CTL model checking problem where whether a configuration of a PDS satisfies an atomic proposition or not depends only on the control state of the configuration. We consider also CTL model checking with regular valuations, where the set of configurations in which an atomic proposition holds is a regular language. We reduce these problems to the emptiness problem in Alternating Büchi Pushdown Systems, and we give an algorithm to solve this emptiness problem. Our algorithms are more efficient than the other existing algorithms for CTL model checking for PDSs in the literature. We implemented our techniques in a tool, and we applied it to different case studies. Our results are encouraging. In particular, we were able to confirm the existence of known bugs in Linux source code.

Model-Checking Linear-Time Properties of Quantum Systems

We define a formal framework for reasoning about linear-time properties of quantum systems in which quantum automata are employed in the modeling of systems and certain (closed) subspaces of state Hilbert spaces are used as the atomic propositions about the behavior of systems. We provide an algorithm for verifying invariants of quantum automata. Then, an automata-based model-checking technique is generalized for the verification of safety properties recognizable by reversible automata and ω--properties recognizable by reversible Büchi automata.

From Games to Truth Functions: A Generalization of Giles’s Game

Motivated by aspects of reasoning in theories of physics, Robin Giles defined a characterization of infinite valued Łukasiewicz logic in terms of a game that combines Lorenzen-style dialogue rules for logical connectives with a scheme for betting on results of dispersive experiments for evaluating atomic propositions. We analyze this game and provide conditions on payoff functions that allow us to extract many-valued truth functions from dialogue rules of a quite general form. Besides finite and infinite valued Łukasiewicz logics, also Meyer and Slaney’s Abelian logic and Cancellative Hoop Logic turn out to be characterizable in this manner.

Domains, plural truth, and mixed atomic propositions

In this paper, I discuss two concerns for pluralist truth theories: a concern about a key detail of these theories and a concern about their viability. The detailed-related concern is that pluralists have relied heavily upon the notion of a domain, but it is not transparent what they take domains to be. Since the notion of a domain has been present in philosophy for some time, it is important for many theorists, not only truth pluralists, to be clear on what domains are and what work they can do. The viability-related concern is that it's not clear how a pluralist truth theory could explain the truth-conditions of mixed atomic propositions. To address this concern, truth pluralists should recognize something to which they have not been sufficiently attentive: that some atomic propositions belong to more than one domain. But, recognizing this requires rethinking the relationships between the nature of propositions, their membership in domains, and their truth. I address these issues and propose an understanding of them that is preferable to the best existing account of them, that offered by Michael Lynch.

A representative model based algorithm for maximal contractions

In this paper, we propose a representative model based algorithm to calculate maximal contractions. For a formal theory Γ and a fact set Δ, the algorithm begins by accepting all models of refutation by facts of Γ with respect to Δ and filters these models to obtain the models of R-refutation. According to the completeness of R-calculus, the relevant maximal contraction is obtained simultaneously. In order to improve the efficiency, we divide the models into different classes according to the assignments of atomic propositions and only select relevant representative models to verify the satisfiability of each proposition. The algorithm is correct, and all maximal contractions of a given problem can be calculated by it. Users could make a selection according to their requirements. A benchmark algorithm is also provided. Experiments show that the algorithm has a good performance on normal revision problems.

Deductive Reasoning in the Structuralist Approach

The distinction between the syntactic and the semantic approach to scientific theories emerged in formal philosophy of science. The semantic approach is commonly considered more advanced and more successful than the syntactic one, but the transition from the one approach to the other was not brought about without any loss. In essence, it is the formal analysis of atomic propositions and the analysis of deductive reasoning that dropped out of consideration in at least some of the elaborated versions of the semantic approach. In structuralist theory of science, as founded by Sneed and Stegmuller, the focus is on global propositions concerning the question of whether or not certain empirical systems satisfy a set-theoretic predicate that encodes the axioms of a scientific theory. Hence, an analysis of deductive reasoning from atomic premisses with the help of a given theory is missing. The objective of the present paper is to develop a deductive system on the basis of the structuralist framework. This system comes with a novel formulation of empirical propositions in structuralism.

An input–output simulation approach to controlling multi-affine systems for linear temporal logic specifications

This article presents an input–output simulation approach to controlling multi-affine systems for linear temporal logic (LTL) specifications, which consists of the following steps. First, the state space is partitioned into rectangles, each of which satisfies atomic LTL propositions. Then, we study the control of multi-affine systems on rectangles, including the control based on the exit sub-region to drive all trajectories starting from a rectangle to exit through a facet and the control to stabilise the multi-affine system towards a desired point. With the proposed controllers, a finitely abstracted transition system is constructed which is shown to be input–output simulated by the rectangular transition system of the multi-affine system. Since the input–output simulation preserves LTL properties, the controller synthesis of the multi-affine system for LTL specifications is achieved by designing a nonblocking supervisor for the abstracted transition system and by implementing the resulting supervisor to the original multi-affine system.

The Uncertainty Relation for Quantum Propositions

Logical propositions with the fuzzy modality "Probably" are shown to obey an uncertainty principle very similar to that of Quantum Optics. In the case of such propositions, the partial truth values are in fact probabilities. The corresponding assertions in the metalanguage, have complex assertion degrees which can be interpreted as probability amplitudes. In the logical case, the uncertainty relation is about the assertion degree, which plays the role of the phase, and the total number of atomic propositions, which plays the role of the number of modes. In analogy with coherent states in quantum physics, we define "quantum coherent propositions" those which minimize the above logical uncertainty relation. Finally, we show that there is only one kind of compound quantum-coherent propositions: the "cat state" propositions.

A Constructive Type-Theoretical Formalism for the Interpretation of Subatomically Sensitive Natural Language Constructions

The analysis of atomic sentences and their subatomic components poses a special problem for proof-theoretic approaches to natural language semantics, as it is far from clear how their semantics could be explained by means of proofs rather than denotations. The paper develops a proof-theoretic semantics for a fragment of English within a type-theoretical formalism that combines subatomic systems for natural deduction [20] with constructive (or Martin-Lof) type theory [8, 9] by stating rules for the formation, introduction, elimination and equality of atomic propositions understood as types (or sets) of subatomic proof-objects. The formalism is extended with dependent types to admit an interpretation of non-atomic sentences. The paper concludes with applications to natural language including internally nested proper names, anaphoric pronouns, simple identity sentences, and intensional transitive verbs.

Robust Vacuity for Branching Temporal Logic

There is a growing interest in techniques for detecting whether a logic specification is satisfied too easily, or vacuously . For example, the specification “every request is eventually followed by an acknowledgment” is satisfied vacuously by a system that never generates any requests. Vacuous satisfaction misleads users of model-checking into thinking that a system is correct. It is a serious problem in practice. There are several existing definitions of vacuity. Originally, Beer et al. [1997] formalized vacuity as insensitivity to syntactic perturbation ( syntactic vacuity ). This formulation captures the intuition of “vacuity” when applied to a single occurrence of a subformula. Armoni et al. argued that vacuity must be robust ; not affected by semantically invariant changes, such as extending a model with additional atomic propositions. They show that syntactic vacuity is not robust for subformulas of linear temporal logic, and propose an alternative definition; trace vacuity . In this article, we continue this line of research. We show that trace vacuity is not robust for branching time logic. We further refine the notion of vacuity so that it applies uniformly to linear and branching time logic and does not suffer from the common pitfalls of prior definitions. Our new definition, bisimulation vacuity , is a proper and nontrivial extension of both syntactic and trace vacuity. We discuss the complexity of detecting bisimulation vacuity, and identify several practically-relevant subsets of CTL* for which vacuity detection problem is reducible to model-checking. We believe that in most practical applications, bisimulation vacuity provides both the desired theoretical properties and is tractable computationally.

Synthesis of Reactive(1) designs

We address the problem of automatically synthesizing digital designs from linear-time specifications. We consider various classes of specifications that can be synthesized with effort quadratic in the number of states of the reactive system, where we measure effort in symbolic steps. The synthesis algorithm is based on a novel type of game called General Reactivity of rank 1 (gr(1)), with a winning condition of the form(□ ◊p1∧⋯∧□ ◊pm)→(□ ◊q1∧⋯∧□ ◊qn), where each pi and qi is a Boolean combination of atomic propositions. We show symbolic algorithms to solve this game, to build a winning strategy and several ways to optimize the winning strategy and to extract a system from it. We also show how to use gr(1) games to solve the synthesis of ltl specifications in many interesting cases. As empirical evidence to the generality and efficiency of our approach we include a significant case study. We describe the formal specifications and the synthesis process applied to a bus arbiter, which is a realistic industrial hardware specification of modest size.

Transparent partial order reduction

Partial Order Reduction (POR) techniques improve the basic model checking algorithm by reducing the numbers of states and transitions explored in verifying a property of the model. In the ample POR framework for the verification of an LTL?X formula ?, one associates to each state s a subset T s of the set of all transitions enabled at s. The approach requires that whenever T s is a proper subset, the transitions in T s must be invisible, i.e., their execution can never change the truth values of the atomic propositions occurring in ?. In this paper, we show that the invisibility restriction can be relaxed: for propositions that only occur negatively in ?, it suffices that the transitions in T s merely never change the truth value from true to false, and for those that occur only positively, from false to true. This opens up opportunities for reduction, in many commonly occurring scenarios, that would not be allowed by the stricter invisibility criterion.

Proposition algebra

Sequential propositional logic deviates from conventional propositional logic by taking into account that during the sequential evaluation of a propositional statement, atomic propositions may yield different Boolean values at repeated occurrences. We introduce “free valuations” to capture this dynamics of a propositional statement's environment. The resulting logic is phrased as an equationally specified algebra rather than in the form of proof rules, and is named “proposition algebra.” It is strictly more general than Boolean algebra to the extent that the classical connectives fail to be expressively complete in the sequential case. The four axioms for free valuation congruence are then combined with other axioms in order define a few more valuation congruences that gradually identify more propositional statements, up to static valuation congruence (which is the setting of conventional propositional logic). Proposition algebra is developed in a fashion similar to the process algebra ACP and the program algebra PGA, via an algebraic specification which has a meaningful initial algebra for which a range of coarser congruences are considered important as well. In addition, infinite objects (i.e., propositional statements, processes and programs respectively) are dealt with by means of an inverse limit construction which allows the transfer of knowledge concerning finite objects to facts about infinite ones while reducing all facts about infinite objects to an infinity of facts about finite ones in return.

Solving conflicts in information merging by a flexible interpretation of atomic propositions

Although many techniques for merging conflicting propositional knowledge bases have already been proposed, most existing work is based on the idea that inconsistency results from the presence of incorrect pieces of information, which should be identified and removed. In contrast, we take the view in this paper that conflicts are often caused by statements that are inaccurate rather than completely false, suggesting to restore consistency by interpreting certain statements in a flexible way, rather than ignoring them completely. In accordance with this view, we propose a novel approach to merging which exploits extra-logical background information about the semantic relatedness of atomic propositions. Several merging operators are presented, which are based on different formalizations of this background knowledge, ranging from purely qualitative approaches, related to possibilistic logic, to quantitative approaches with a probabilistic flavor. Both syntactic and semantic characterizations are provided for each merging operator, and the computational complexity is analyzed.

Gappy Propositions?

After introducing Millianism and touching on two problems raised by genuinely empty names for Millianism (section I), I provide a brief exposition of the Gappy Proposition View (GPV) and of how different versions of this view can reply to the problems in question (section II). In the following sections I develop my reasons against the GPV. First, I will try to argue that apparently promising arguments for the claim that gappy propositions are propositions are not successful (section III). Then, I will develop two arguments against GPs via demonstrating two odd consequences of the GPV: (a) that there can be an atomic proposition which contains other propositions that are not the semantic contents of any part of the sentence expressing that atomic proposition, and (b) that propositional structures are propositions (section IV). And finally, I will attempt to show that if any of these views can provide a successful defense of Millianism, it can do so without GPs, given some slight changes (section V). I will conclude that GPs should be avoided (section VI).

Metric propositional neighborhood logics on natural numbers

Interval logics formalize temporal reasoning on interval structures over linearly (or partially) ordered domains, where time intervals are the primitive ontological entities and truth of formulae is defined relative to time intervals, rather than time points. In this paper, we introduce and study Metric Propositional Neighborhood Logic (MPNL) over natural numbers. MPNL features two modalities referring, respectively, to an interval that is "met by" the current one and to an interval that "meets" the current one, plus an infinite set of length constraints, regarded as atomic propositions, to constrain the length of intervals. We argue that MPNL can be successfully used in different areas of computer science to combine qualitative and quantitative interval temporal reasoning, thus providing a viable alternative to well-established logical frameworks such as Duration Calculus. We show that MPNL is decidable in double exponential time and expressively complete with respect to a well-defined sub-fragment of the two-variable fragment $${{\rm FO}^2[\mathbb{N},=,<,s]}$$ of first-order logic for linear orders with successor function, interpreted over natural numbers. Moreover, we show that MPNL can be extended in a natural way to cover full $${{\rm FO}^2[\mathbb{N},=,<,s]}$$ , but, unexpectedly, the latter (and hence the former) turns out to be undecidable.

What Tipper is Ready for: A Semantics for Incomplete Predicates

Abstract This paper presents a precise semantics for incomplete predicates such as “ready”. Incomplete predicates have distinctive logical properties that a semantic theory needs to accommodate. For instance, “Tipper is ready” logically implies “Tipper is ready for something”, but “Tipper is ready for something” does not imply “Tipper is ready”. It is shown that several approaches to the semantics of incomplete predicates fail to accommodate these logical properties. The account offered here defines contexts as structures containing an element called a proposition set, which contains atomic propositions and negations of atomic propositions. The condition under which “Tipper is ready” is true in a context is defined in terms of the contents of the proposition set for the context. On this account, the content of the context pertinent to a conversation must be determined not by what speakers have in mind but by relations of objective relevance.

An Inconsistency-Tolerant Approach to Information Merging Based on Proposition Relaxation

Inconsistencies between different information sources may arise because of statements that are inaccurate, albeit not completely false. In such scenarios, the most natural way to restore consistency is often to interpret assertions in a more flexible way, i.e. to enlarge (or relax) their meaning. As this process inherently requires extra-logical information about the meaning of atoms, extensions of classical merging operators are needed. In this paper, we introduce syntactic merging operators, based on possibilistic logic, which employ background knowledge about the similarity of atomic propositions to appropriately relax propositional statements.